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Generalized derivations with skew nilpotent values on Lie ideals
Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)n =(ax)n for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal and a noncommutative Lie ideal L, respectively.
Let R be a prime ring with extended centroid C, a nonzero ideal I and two derivations d1, d2. Suppose that d1(x)ⁿ = d2(x)ⁿ for all x ∈ I. Then there exists λ ∈ C such that d2(x) = λd1(x) for all x ∈ R.
Let R be a ring with a subset S. A mapping of R into itself is called strong commutativity-preserving (scp) on S, if [f(x), f(y)] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered. © 2008 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH.
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